Combine the egg yolks, mustard, a tablespoon of the lemon juice and the garlic in a blender or food processor. Blend at medium speed for about 45 seconds.
While the blender is running, add the oil a few drops at a time until the contents of the container noticeably begin to change texture. Then step up the addition of the oil to a thin stream while the blender continues to run. When all the oil is incorporated, add the rest of the lemon juice, salt, and cayenne. Refrigerate to thicken.